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We study distributional solutions of semilinear biharmonic equations of the typeΔ  2   u  +  f  (  u  )  =  0     on   ℝ  N   ,{\Delta ^2}u + f(u) = 0\quad on\;{{\mathbb R} ^N},where  f  is a continuous function satisfying  f  ( t ) t  ≥  c  | t |  q +1  for all  t  ∈ ℝ with  c  &gt; 0 and  q  &gt; 1. By using a new approach mainly based on careful choice of suitable weighted test functions and a new version of Hardy- Rellich inequalities, we prove several Liouville theorems independently of the dimension  N  and on the sign of the solutions.

Entire solutions of certain fourth order elliptic problems and related inequalities